FRG: Collaborative Research: Chern classes in Iwasawa Theory

迄今为止已研究的岩泽理论的主要猜想将第一类岩泽模和Selmer复形与p进L级联系起来。该 FRG 项目的目标是将这一理论推广到更高的陈省身类别。这一概括的一个组成部分涉及如何以一种有利于通过 L 级数研究它们的方式来定义更高的 Chern 类。这将通过将 Parshin 和 Beilinson 的 adelic 方法扩展到 Iwasawa 理论的背景来完成。泛化的另一个组成部分与将更高的 Chern 类不变量连接到 L 级数有关。 为此，算术问题需要足够的结构才能使用 L 级数了解其更高的余维特征。将考虑三种特殊情况：(i) 格林伯格对全实域的猜想，(ii) 分裂素数处虚二次域的岩泽理论，以及 (iii) 函数域情况。 关于(i)，格林伯格推测自然岩泽模在余维一中具有微不足道的支持； PI 将使用 L 系列研究他们的余维二支持。关于 (ii)，Rubin、Kings 和 Johnson-Leung 的工作表明，应该通过与 p 进 L 级数对相关的 K_2 群中的符号来研究第二陈类。 关于 (iii)，PI 将研究由 Witte 在 Iwasawa 代数的较高相关 K 群内的函数域情况下定义的类的 Chern 类映射下的图像。 该项目的另一个组成部分与将证明第一陈类主要猜想所使用的归约技术推广到更高的陈类有关。这涉及将倾斜复数理论和派生等价推广到岩泽代数，该理论用于群表示理论和研究傅立叶-向井函子。该提案涉及有关代数方程对称群的基本问题。 20 世纪 50 年代，岩泽开始采用一种新方法来研究此类方程，考虑它们在无限家庭中的行为。岩泽表明，许多这样的家庭都有明确的渐近行为。这导致了关于此类族中对称群的数值增长率的基本猜想。 证明此类“主要猜想”一直是过去 50 年来抽象代数的中心目标之一。 该提议与这些猜想的改进有关，这些猜想涉及更精确的增长率测量。 就广泛的影响而言，此类代数问题的研究促进了对社会至关重要的技术的发展，例如改进的数据压缩和安全传输。

The Main Conjectures of Iwasawa theory which have been studied up to now relate the first Chern classes of Iwasawa modules and Selmer complexes to p-adic L-series. The object of this FRG project is to generalize this theory to higher Chern classes. One component of this generalization concerns how to define higher Chern classes in a way that facilitates studying them by L-series. This will be done by extending to the context of Iwasawa theory the adelic methods of Parshin and Beilinson. Another component of the generalization has to do with connecting higher Chern class invariants to L-series. To do this, one needs enough structure in the arithmetic problem to see into its higher codimension features using L-series. Three particular cases will be considered are (i) Greenberg's conjecture over totally real fields, (ii) Iwasawa theory for imaginary quadratic fields at split primes, and (iii) the function field case. Concerning (i), Greenberg has conjectured that the natural Iwasawa modules have trivial support in codimension one; the PIs will study their codimension two support using L-series. Concerning (ii), work of Rubin, and of Kings and Johnson-Leung, suggests that one should study second Chern classes via symbols in K_2 groups associated to pairs of p-adic L-series. Concerning (iii), the PIs will study the images under Chern class maps of classes defined by Witte in the function field case inside the higher relative K-groups of Iwasawa algebras. One further component of this project has to do with generalizing to higher Chern classes the reduction techniques used in proving first Chern class Main Conjectures. This involves generalizing to Iwasawa algebras the theory of tilting complexes and derived equivalences which is used in group representation theory and in studying Fourier-Mukai functors.This proposal deals with fundamental questions about the groups of symmetries of algebraic equations. In the 1950's, Iwasawa began a new approach to the study of such equations by considering their behavior in infinite families. Iwasawa showed that many such families have well defined asymptotic behavior. This led to fundamental conjectures concerning the numerical growth rate of the symmetry groups arising from such families. The proof of such "Main Conjectures" has been one of the central goals of abstract algebra over the last 50 years. This proposal has to do with the refinements of these conjectures which deal with more precise measures of rates of growth. Concerning broad impacts, work on algebraic questions of this kind has led to the development of technology essential to society, such as the improved compression and secure transmission of data.